On the Best Approximation in the Mean by Algebraic Polynomials with Weight and the Exact Values of Widths for the Classes of Functions

is obtained on the class L2r(Dρ), where r ∈ ℤ+; \( {D}_{\rho} = \sigma (x)\frac{d^2}{d{ x}^2}+\tau (x)\frac{d}{d x} \), σ and τ are polynomials of at most the second and first degrees, respectively, ρ is a weight function, 0 < p ≤ 2, 0 < h < 1, λn(ρ)
are eigenvalues of the operator Dρ, φ
is a nonnegative measurable and summable function (in the interval (a, b)) which is not equivalent to zero, Ωk,ρ is the generalized modulus of continuity of the k th order in the space L2,ρ (a, b), and En (f)2,ρ
is the best polynomial approximation in the mean with weight ρ for a function f ∈ L2,ρ (a, b). The exact values of widths for the classes of functions specified by the characteristic of smoothness Ωk,ρ and the K-functional \( \mathbb{K} \)m are also obtained.